1. Statement of the Technical Field
The invention concerns numerical sequence generation. More particularly, the invention concerns a method for digitally generating a chaotic numerical sequence.
2. Description of the Related Art
Chaotic systems can generally be thought of as systems which vary unpredictably due to the defining characteristics of: sensitivity to initial conditions; being dense; and being topological transitive. The characteristics of denseness and topological transitivity mean that the resultant numerical values generated by a chaotic circuit do not clump together. When measured or observed, chaotic systems do not reveal any discernible regularity or order. Chaotic systems are distinguished by a sensitive dependence on a set of initial conditions and by having an evolution through time and space that appears to be quite random. However, despite its “random” appearance, chaos is a deterministic evolution.
There are many types of chaotic communications systems known in the art. Such chaotic communications systems include a chaotic spreading based communications system, a coherent chaos shift keying based communications system, a non-coherent chaos shift keying based communications system, and a differential code shift keying (DCSK) based communications system. Such chaotic communications systems also include a chaotic on-off keying based communications system, a frequency-modulated DCSK based communications system, a correlation delay shift keying based communications system, a symmetric code shift keying (CSK) based communications system, and a quadrature CSK based communications system.
Chaotic communications systems offer promise for being the basis of a next generation of low probability of intercept (LPI) waveforms, low probability of detection (LPD) waveforms, and secure waveforms. White many chaotic communications systems have been developed for generating chaotically modulated waveforms, such chaotic communications systems suffer from low throughput. The term “throughput” as used herein refers to the amount of payload data transmitted over a data link during a specific amount of time.
The throughput limitation with chaos based communication systems can be traced to the way in which chaos generators have been implemented. Chaos generators have been conventionally constructed using analog chaotic circuits. The reason for reliance on analog circuits for this task has been the widely held conventional belief that efficient digital generation of chaos is impossible. Notwithstanding the apparent necessity of using analog type chaos generators, that approach has not been without problems. For example, analog chaos generator circuits are known to drift over time. The term “drift” as used herein refers to a slow long term variation in one or more parameters of a chaotic signal. Another problem with such analog circuits is that state information must be constantly transferred over a communication channel to keep a transmitter and receiver synchronized, thereby reducing data throughput. Still another problem with analog chaotic circuits is that their rate of advancement from state to state is fixed by the physical characteristics of the circuit and is not easily verified.
The transmitter and receiver in chaos based communication systems are often synchronized by exchanging state information over a data link. Such synchronization can reduce the problems associated with drift by communicating updated state information. However, such a synchronization process offers diminishing return because state information must be exchanged more often between the transmitter and the receiver to obtain a high data rate. This high data rate results in a faster relative drift. In effect, state information must be exchanged at an increased rate between the transmitter and receiver to counteract the faster relative drift. Although some chaotic communications systems employ a relatively efficient synchronization process, these chaotic communications systems still suffer from low throughput.
Communications systems may use multiple pseudo-random number generators to generate a digital chaotic-like sequence. However, such communications systems only produce more complex pseudo-random number sequences that possess all pseudo-random artifacts and no true chaotic properties. While certain polynomials can mimic chaotic behavior, the arithmetic precision required to generate chaotic number sequences requires an impractical implementation. Stated differently, the binary arithmetic necessary in order to achieve digital chaos is prohibitive.
In view of the forgoing, there is a need for a chaotic communications system having an increased throughput. There is also a need for a chaotic communications system configured for generating a signal having chaotic properties. As such, there is further a need for a chaotic communications system that offers state drift and update properties that are more favorable to high data rate applications. Most significantly, there is a need for a method for digitally generating a chaotic number sequence that can be used for a variety of communications system applications.